Evolution and Optimum Seeking

378 Appendix B
man), vol. 15 of Problemata series, Verlag Frommann-Holzboog, Stuttgart, 1973 also
H.-P. Schwefel, Numerical Optimization of Computer Models, Wiley, Chichester, 1981
(translated by M. W. Finnis from Numerische Optimierung von Computer-Modellen mittels
der Evolutionsstrategie, vol. 26 of Interdisciplinary Systems Research, Birkhauser,
Basle, Switzerland, 1977).
The current L parameter vectors are used to generate LL new ones by means of small
random changes.
The best L of these become the initial ones for the next generation (iteration). At the
same time, the step sizes (standard deviations) for the changes in the variables (strategy
parameters) are altered. The selection leads to adaptation to the local topology if LL/L
is assigned a suitably large value, e.g., >6. The random changes in the parameters are
produced by the addition of normally distributed random numbers, while those in the
step sizes are produced from random numbers with a log-normal distribution by multiplication.
4. Convergence criterion
Based on the di erences in value of the objective function (see under EC and ED).
5. Peripheral I/O: none.
6. Notes
The multimembered strategy represents an improvement in reliabilityover the two membered
strategy. On the other hand, the run time is greater when an ordinary (serial)
digital computer is used. The run time increases less rapidly than in proportion to LL
(the number of descendants per generation), because increasing LL increases the convergence
rate (over the generations). However, minima at a boundary of the feasible
region or at a vertex are attained only slowly or inexactly. In any case, although the
certainty of global convergence cannot be guaranteed, numerical tests have shown that
the multimembered strategy is far better than other search procedures in this respect. It
is capable of handling separated feasible regions provided that the number of parameters
is not large and that the initial step sizes are set suitably large. In doubtful cases it is
recommended to repeat the search each time with a di erent set of initial values and/or
random numbers. If the optimum being sought lies at a boundary of the feasible region, it
is probably better to choose a value for SN (the parameter governing the rates of change
of the standard deviations) less than the (maximal) value suggested above.
7. Subroutines or functions used
The function names are to be declared as external in the segment that calls GRUP.
7.1 Objective function
To be written by the user in the form: